S-parameter measurements using real-time oscilloscopes

ABSTRACT

A method for determining scattering parameters of a device under test using a real-time oscilloscope. The method includes calculating a reflection coefficient of each port of a device under test with N ports, wherein N is greater than one, based on a first voltage measured by the real-time oscilloscope when a signal is generated from a signal generator. The method also includes determining an insertion loss coefficient of each port of the device under test, including calculating the insertion loss coefficient of the port of the device under test to be measured based on a second voltage measured by the real-time oscilloscope when a signal is generated from a signal generator.

BENEFIT

This application claims benefit of U.S. Provisional Application No.61/026,434, filed Jul. 18, 2014, titled S-PARAMETER MEASUREMENTS USINGREAL-TIME OSCILLOSCOPES, which is incorporated herein by reference inits entirety.

TECHNICAL FIELD

This disclosure relates to a method for measuring complete two-port andmulti-port scattering parameters (S-parameters) of a device under test(DUT) using a real-time oscilloscope, a power divider and a signalsource.

BACKGROUND

Traditionally a vector network analyzer or a time-domain reflectometer(TDR) system with a sampling oscilloscope has been required to obtainS-parameter measurements for characterizations of a device under test(DUT). If a fixture is used to measure the S-parameters, then once theS-parameters of the fixture have been measured and the S-parameters ofthe DUT plus the fixture have been measured, a full de-embed operationcan be performed to obtain only characteristics of the DUT by taking outthe errors due to the fixture. However, vector network analyzers and TDRsystems are expensive.

As bit rate goes higher, high speed serial data link simulation andmeasurements require using S-parameters of the DUT to characterize thecomponents within the simulated link for embedding and de-embeddingoperations. For example, as seen in FIG. 1, the output impedance of thetransmitter 100, the input impedance of the receiver 102, and the fullS-parameters of the channel 104 are all needed to fully characterize andsimulate the link so that accurate test and measurements of the DUT maybe made.

Real-time oscilloscopes are widely used to do high speed serial datalink debugging, testing, and measurements. It is desired to use thereal-time oscilloscopes to also measure the S-parameters of the DUT andthen use the measured S-parameters for other measurements andsimulations, without having to use multiple instruments.

Embodiments of the disclosed technology address these and otherlimitations in the prior art.

SUMMARY

Certain embodiments of the disclosed technology include a method fordetermining scattering parameters of a device under test using areal-time oscilloscope. The method includes determining a reflectioncoefficient of each port of a device under test with N ports, wherein Nis an integer greater than or equal to one, including terminating theports of the device under test not being measured with a resistor, andcalculating the reflection coefficient of the port of the device undertest to be measured based on a first voltage measured by the real-timeoscilloscope when a signal is generated from a signal generator. Themethod also includes determining an insertion loss and/or crosstalkcoefficient between two different ports of the device under test,including calculating the insertion loss and/or a crosstalk between theports of the device under test to be measured based on a second voltagemeasured by the real-time oscilloscope when a signal is generated from asignal generator.

Certain embodiments of the disclosed technology also include a system,comprising a power divider, a signal generator in communication with thepower divider, the real-time oscilloscope and/or a device under test,the signal generator configured to generate a signal, and a real-timeoscilloscope in communication with the signal generator, the powerdivider and the device under test. The real-time oscilloscope isconfigured to calculate a reflection coefficient of each port of adevice under test with N ports, wherein N is greater than or equal toone, measured based on a first voltage measured by the real-timeoscilloscope when a signal is generated from a signal generator, andcalculating an insertion loss and/or a crosstalk between the ports ofthe device under test to be measured based on a second voltage measuredby the real-time oscilloscope when a signal is generated from a signalgenerator. A synchronized trigger between the signal generator and thereal-time oscilloscope provides an absolute time reference for returnloss coefficient, insertion loss and cross talk measurements.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a serial data link system.

FIG. 2 is a block diagram of a return loss coefficient measurementsystem according to embodiments of the disclosed technology.

FIG. 3 is a signal flow diagram of the system shown in FIG. 2.

FIG. 4 is a block diagram of an insertion loss measurement systemaccording to embodiments of the disclosed technology.

FIG. 5 is a signal flow diagram of the system shown in FIG. 4.

FIG. 6 is a power divider of FIG. 2.

DETAILED DESCRIPTION

In the drawings, which are not necessarily to scale, like orcorresponding elements of the disclosed systems and methods are denotedby the same reference numerals.

The disclosed technology measures S-parameters of a DUT 206 using areal-time oscilloscope 200, a signal generator 202, and a power divider204, as shown in FIG. 2. A two-port DUT 206 has four S-parameters:reflection coefficients s₁₁ ^(DUT), s₂₂ ^(DUT) and insertion loss and/orcrosstalk terms s₁₂ ^(DUT), s₂₁ ^(DUT).

In the setup shown in FIG. 2, the synchronized trigger 208 between thesignal generator 202 and the real-time oscilloscope 200 provides anabsolute time reference for the reflection coefficients and theinsertion loss and/or crosstalk coefficients. This synchronizing actionestablishes a zero time or zero phase reference point for theacquisition system of the oscilloscope 200. Signal generator 202 mayeither output sweep sine signals or fast step signals that may cover awide bandwidth. For example, a fast step signal can cover more than 50GHz bandwidth. For swept sine signals, zero phase is the referencepoint. For fast step signals, zero time is the reference point.

Power divider 204 may be any type of power divider or splitter. Forexample, power divider 204 may be a power divider having three 16⅔ Ohmresistors at three branches creating a three way power network, as seenin FIG. 6.

FIG. 3 illustrates a signal flow diagram of the system shown in FIG. 2.

The power divider 204, as seen in the signal flow diagram of FIG. 3, isa three-port network and characterized by a three port S-parameters dataset:

$\begin{matrix}{s = \begin{bmatrix}s_{11} & s_{12} & s_{13} \\s_{21} & s_{22} & s_{23} \\s_{31} & s_{32} & s_{33}\end{bmatrix}} & (1)\end{matrix}$

The S-parameters of the power divider 204 may be measured ahead of timeand saved in a memory (not shown) of the real-time oscilloscope 200. TheDUT 206 is modeled as a two port S-parameters data set s^(DUT).

To measure the reflection coefficient s₁₁ ^(DUT) of port one 210 of theDUT 206, port two 212 of the DUT 206 is terminated with an ideal 50 Ohmresistor 214. The signal generator 202 is modeled with its ideal voltageν_(s) and has the reflection coefficient s₂₂ ^(ss) at the output port.The real-time oscilloscope 200 has the reflection coefficient s₁₁^(scope) at its input port.

As the power divider is a three-port network, with port two beingterminated by DUT 206, a two-port system can be derived from port oneand port three of the power divider 204 as follows:

${(2)\begin{bmatrix}b_{1} \\b_{2}\end{bmatrix}} = {\left\{ {\begin{bmatrix}s_{11} & s_{12} \\s_{21} & s_{22}\end{bmatrix} + {\begin{bmatrix}s_{13} \\s_{23}\end{bmatrix}{{s_{11}^{DUT}\left\lbrack {1 - {s_{33}s_{11}^{DUT}}} \right\rbrack}\begin{bmatrix}s_{31} & s_{32}\end{bmatrix}}}} \right\} \begin{bmatrix}a_{1} \\a_{2}\end{bmatrix}}$

Let equation (3):

$\begin{matrix}\begin{matrix}{\overset{\sim}{s} = \begin{bmatrix}{\overset{\sim}{s}}_{11} & {\overset{\sim}{s}}_{12} \\{\overset{\sim}{s}}_{21} & {\overset{\sim}{s}}_{22}\end{bmatrix}} \\{= {\begin{bmatrix}s_{11} & s_{12} \\s_{21} & s_{22}\end{bmatrix} + {\begin{bmatrix}s_{13} \\s_{23}\end{bmatrix}{{s_{11}^{DUT}\left\lbrack {1 - {s_{33}s_{11}^{DUT}}} \right\rbrack}^{- 1}\begin{bmatrix}s_{31} & s_{32}\end{bmatrix}}}}}\end{matrix} & (3)\end{matrix}$

denote equation (4):

{tilde over (s)} ₁₁ ^(DUT) =s ₁₁ ^(DUT)[1−s ₃₃ s ₁₁ ^(DUT)]⁻¹  (4)

Then equation (3) can be re-written as:

$\begin{matrix}{\begin{bmatrix}{\overset{\sim}{s}}_{11} & {\overset{\sim}{s}}_{12} \\{\overset{\sim}{s}}_{21} & {\overset{\sim}{s}}_{22}\end{bmatrix} = \begin{bmatrix}{s_{11} + {s_{13}{\overset{\sim}{s}}_{11}^{DUT}s_{31}}} & {s_{12} + {s_{13}{\overset{\sim}{s}}_{11}^{DUT}s_{32}}} \\{s_{21} + {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{31}}} & {s_{22} + {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{32}}}\end{bmatrix}} & (5)\end{matrix}$

From equation (2), the transfer function from the voltage source of thesignal generator 202 to the input to the real-time oscilloscope can bederived as:

$\begin{matrix}{\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{1 - s_{22}^{ss}}{{\left( {1 - {s_{22}^{ss}{\overset{\sim}{s}}_{11}}} \right){{\overset{\sim}{s}}_{21}^{- 1}\left( {1 - {{\overset{\sim}{s}}_{22}s_{11}^{scope}}} \right)}} - {s_{22}^{ss}{\overset{\sim}{s}}_{12}s_{11}^{scope}}}}} & (6)\end{matrix}$

The terms of equation (5) may be inserted into equation (6), andequation (6) can be rewritten as:

$\begin{matrix}{\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{\left( {1 - s_{22}^{ss}} \right)\left( {s_{21} + {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{31}}} \right)}{\begin{matrix}{1 - {s_{22}^{ss}\left( {s_{11} + {s_{12}{\overset{\sim}{s}}_{11}^{DUT}s_{31}}} \right)} - {\left( {s_{22} + {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{32}}} \right)s_{11}^{scope}} +} \\{\begin{bmatrix}{\left( {{s_{11}s_{22}} - {s_{21}s_{12}}} \right) +} \\{\left( {{s_{13}s_{31}s_{22}} + {s_{23}s_{32}s_{11}} - {s_{23}s_{31}s_{12}} + {s_{13}s_{32}s_{21}}} \right){\overset{\sim}{s}}_{11}^{DUT}}\end{bmatrix}s_{22}^{ss}s_{11}^{scope}}\end{matrix}}}} & (7)\end{matrix}$

In equation (7), ν_(s) is the voltage source value of the signalgenerator 202 and may be obtained during a calibration process and savedin the memory of the real-time oscilloscope 200. b₂ is the voltageacquired by the oscilloscope 200 when ν_(s) is generated, so it isknown. As discussed above, all the S-parameter terms for the powerdivider 204 may be measured ahead of time, and stored in the memory ofthe oscilloscope 200. s₂₂ ^(ss), corresponding to the signal sourceoutput impedance, may also be measured ahead of time and stored inmemory. s₁₁ ^(scope), corresponding to the real-time oscilloscope 200input impedance, may be measured ahead of time and stored in the memoryas well. The only unknown, therefore, is {tilde over (s)}₁₁ ^(DUT) whichmay be computed from equation (7). Once {tilde over (s)}₁₁ ^(DUT) isobtained from equation (7), s₁₁ ^(DUT), the reflection coefficient ofthe DUT 206 may be calculated using equation (4) above.

Equation (7) may be simplified with some assumptions. First, theS-parameters of an ideal power divider is:

$\begin{matrix}{s = {\begin{bmatrix}s_{11} & s_{12} & s_{13} \\s_{21} & s_{22} & s_{23} \\s_{31} & s_{32} & s_{33}\end{bmatrix} = \begin{bmatrix}0 & \frac{1}{2} & \frac{1}{2} \\\frac{1}{2} & 0 & \frac{1}{2} \\\frac{1}{2} & \frac{1}{2} & 0\end{bmatrix}}} & (8)\end{matrix}$

For an ideal signal generator and an ideal real-time oscilloscope, thesignal source impedance and oscilloscope input impedance are zero:

s₂₂ ^(ss)=0, s₁₁ ^(scope)=0.  (9)

Assuming the S-parameters of the power divider 204, signal generator202, and real-time oscilloscope 200 are their ideal models, thenequation (7) can be simplified as:

$\begin{matrix}{\frac{b_{2}}{v_{s}} = {\frac{1}{4}\left( {1 + {\frac{1}{2}s_{11}^{DUT}}} \right)}} & (10)\end{matrix}$

However, in the real world, the S-parameters of the power divider 204,signal generator 202 and real-time oscilloscope 200 will not be ideal,but the following may be assumed to be true:

i s₂₂ ^(ss) s ₁₁ ^(scope)<<1, s ₂₂ ^(ss) s ₁₁<<1, s ₂₂ s ₁₁^(scope)<<1  (11)

Then equation (7) can be approximated as:

$\begin{matrix}{\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{\left( {1 - s_{22}^{ss}} \right)\left( {s_{21} + {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{31}}} \right)}{1 - {s_{22}^{ss}s_{12}{\overset{\sim}{s}}_{11}^{DUT}s_{31}} - {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{32}s_{11}^{scope}}}}} & (12)\end{matrix}$

Equation (12) provides a working practical approximation of thereflection coefficient of the DUT 206.

To measure s₂₂ ^(DUT), the port two reflection coefficient of the DUT206, port two 212 of DUT 206 is connected to port one of the powerdivider 202 and port one 210 of the DUT 206 is terminated with an ideal50 Ohm resistor 214. Then, the same procedure used to measure the portone 210 reflection coefficient s₁₁ ^(DUT) discussed above may berepeated to calculate the port two 212 reflection coefficient s₂₂^(DUT).

The insertion loss and/or crosstalk terms s₁₂ ^(DUT), s₂₁ ^(DUT) may bemeasured using the system shown in FIG. 4. This system includes thesignal generator 202, the DUT 206, the real-time oscilloscope 200 andthe synchronized trigger 208. The synchronized trigger 208 provides thetime reference so the phases of the measured S-parameters are correct.The signal flow diagram of the system in FIG. 4 can be seen in FIG. 5.

The transfer function from the signal generator's 202 voltage source tothe input to the real-time oscilloscope can be derived in the same wayas equation (6):

$\begin{matrix}{\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{1 - s_{22}^{ss}}{{\left( {1 - {s_{22}^{ss}s_{11}^{DUT}}} \right){s_{11}^{{DUT} - 1}\left( {1 - {s_{22}^{DUT}s_{11}^{scope}}} \right)}} - {s_{22}^{ss}s_{12}^{DUT}s_{11}^{scope}}}}} & (13)\end{matrix}$

Using the assumptions of equation (11), equation (13) can beapproximated as

$\begin{matrix}{\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{\left( {1 - s_{22}^{ss}} \right)s_{21}^{DUT}}{1 - {s_{22}^{ss}s_{11}^{DUT}} - {s_{22}^{DUT}s_{11}^{scope}}}}} & (14)\end{matrix}$

Since s₁₁ ^(DUT) and s₂₂ ^(DUT) have been calculated as discussed above,there is only one unknown variable, s₂₁ ^(DUT), in equation (14).Therefore, equation (14) can be solved for s₂₁ ^(DUT). b₂ in equation(14) is the voltage acquired by the oscilloscope 200 when ν_(s) isgenerated when the devices are configured as in FIG. 4.

To measure s₁₂ ^(DUT), port two of DUT 206 is connected to the signalgenerator 202, port one of DUT 206 is connected to the real-timeoscilloscope 200, and the same procedure that measures s₂₁ ^(DUT) may berepeated to measure s₁₂ ^(DUT).

The method described above may be expanded to measure the reflectioncoefficients and insertion loss and crosstalk terms for a multi-portDUT. To measure an N-port network, the reflection coefficients can bemeasured and calculated by terminating all the other ports of the DUT,except the port connected to the power divider, with an ideal 50 Ohms,using the method described above. Then, the insertion loss and crosstalkterms may be measured by terminating all other ports except two portsconnected to the signal generator and to the real-time oscilloscope,using the above-discussed insertion loss and crosstalk measurementmethod.

Having described and illustrated the principles of the disclosedtechnology in a preferred embodiment thereof, it should be apparent thatthe disclosed technology can be modified in arrangement and detailwithout departing from such principles. We claim all modifications andvariations coming within the spirit and scope of the following claims.

1. A method for determining scattering parameters of a device under testusing a real-time oscilloscope, comprising: determining a reflectioncoefficient of each port of a device under test with N ports, wherein Nis an integer greater than or equal to one, including: terminating theports of the device under test not being measured with a resistor, andcalculating the reflection coefficient of the port of the device undertest based on a first voltage measured by the digital oscilloscope whena signal is generated from a signal generator; and determining aninsertion loss coefficient of each port of the device under test,including calculating the insertion loss coefficient of the port of thedevice under test based on a second voltage measured by the digitaloscilloscope when a signal is generated from a signal generator; andproviding an absolute time reference between the signal generator andthe digital oscilloscope through a synchronized trigger.
 2. The methodof claim 1, further comprising: sending the signal generated from thesignal generator to a first port of a power divider; and measuring thefirst voltage by the real-time oscilloscope at a second port of thepower divider based on the generated signal while the device under testis connected to a third port of the power divider.
 3. The method ofclaim 2, wherein N is two, the method further comprising: sending thesignal generated from the signal generator to the port of the deviceunder test not being measured and to the real-time oscilloscope; andmeasuring the second voltage by the real-time oscilloscope at the portof the device under test being measured based on the generated signal.4. The method of claim 2, wherein N is greater than two, the methodfurther comprising: sending the signal generated from the signalgenerator to one port of the device under test not being measured and tothe real-time oscilloscope; terminating the remaining ports of thedevice under test not being measured with a resistor; and measuring thesecond voltage by the real-time oscilloscope at the port of the deviceunder test being measured based on the generated signal.
 5. The methodof claim 1, wherein the resistor is a 50 Ohm resistor.
 6. The method ofclaim 2, wherein the reflection coefficient is calculated using theequations:$\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{\left( {1 - s_{22}^{ss}} \right)\left( {s_{21} + {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{31}}} \right)}{1 - {s_{22}^{ss}s_{12}{\overset{\sim}{s}}_{11}^{DUT}s_{31}} - {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{32}s_{11}^{scope}}}}$and${\overset{\sim}{s}}_{11}^{DUT} = {s_{11}^{DUT}\left\lbrack {1 - {s_{33}s_{11}^{DUT}}} \right\rbrack}^{- 1}$where b₂ is the first voltage, ν_(s) is the generated signal, s₂₂ ^(ss)is an impedance of the signal generator, s₂₁, s₁₂, S₃₁, S₂₃, S₃₂, S₃₃are the scatter parameter terms of the power divider, s₁₁ ^(scope) is aninput impedance of the real-time oscilloscope, and s₁₁ ^(DUT) is thereflection coefficient.
 7. The method of claim 3, wherein N is two andthe reflection coefficients are calculated for each port using theequations:$\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{\left( {1 - s_{22}^{ss}} \right)\left( {s_{21} + {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{31}}} \right)}{1 - {s_{22}^{ss}s_{12}{\overset{\sim}{s}}_{11}^{DUT}s_{31}} - {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{32}s_{11}^{scope}}}}$and${\overset{\sim}{s}}_{11}^{DUT} = {s_{11}^{DUT}\left\lbrack {1 - {s_{33}s_{11}^{DUT}}} \right\rbrack}^{- 1}$where b₂ is the first voltage, ν_(s) is the generated signal, s₂₂ ^(ss)is an impedance of the signal generator, s₂₁, s₁₂, s₃₁, s₂₃, s₃₂, s₃₃are the scatter parameter terms of the power divider, s₁₁ ^(scope) is aninput impedance of the real-time oscilloscope, and s₁₁ ^(DUT) is thereflection coefficient.
 8. The method of claim 7, wherein the calculatedreflection coefficient of the first port of the device under test is s₁₁^(DUT) and the calculated reflection coefficient of the second port ofthe device under test is s₁₁ ^(DUT), and wherein the insertion losscoefficient is calculated for each port using the equation:$\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{\left( {1 - s_{22}^{ss}} \right)s_{21}^{DUT}}{1 - {s_{22}^{ss}s_{11}^{DUT}} - {s_{22}^{DUT}s_{11}^{scope}}}}$where s₂₁ ^(DUT) is the insertion loss coefficient.
 9. A test andmeasurement system, comprising: a power divider; a signal generator incommunication with the power divider or a device under test, the signalgenerator configured to generate a signal; and a real-time oscilloscopein communication with the signal generator, the power divider and thedevice under test, the real-time oscilloscope configured to: calculate areflection coefficient of each port of a device under test with N ports,wherein N is greater than one, measured based on a first voltagemeasured by the real-time oscilloscope when a signal is generated from asignal generator, and calculating an insertion loss coefficient of theport of the device under test to be measured based on a second voltagemeasured by the real-time oscilloscope when a signal is generated from asignal generator.
 10. The system of claim 9, further comprising asynchronized trigger configured to provide an absolute time referencebetween the signal generator and the real-time oscilloscope.
 11. Thesystem of claim 9, wherein the real-time oscilloscope is furtherconfigured to measure the first voltage by the real-time oscilloscope ata second port of the power divider based on a generated signal while thedevice under test is connected to a third port of the power divider. 12.The system of claim 11, wherein N is two, wherein the signal generatedfrom the signal generator is sent to the port of the device under testnot being measured and to the real-time oscilloscope; and wherein thereal-time oscilloscope is further configured to measure the secondvoltage by the real-time oscilloscope at the port of the device undertest being measured based on the generated signal.
 13. The system ofclaim 12, wherein N is two and the real-time oscilloscope is furtherconfigured to calculate the reflection coefficients for each port usingthe equations:$\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{\left( {1 - s_{22}^{ss}} \right)\left( {s_{21} + {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{31}}} \right)}{1 - {s_{22}^{ss}s_{12}{\overset{\sim}{s}}_{11}^{DUT}s_{31}} - {s_{23}{\overset{\sim}{s}}_{11}^{DUT}s_{32}s_{11}^{scope}}}}$and${\overset{\sim}{s}}_{11}^{DUT} = {s_{11}^{DUT}\left\lbrack {1 - {s_{33}s_{11}^{DUT}}} \right\rbrack}^{- 1}$where b₂ is the first voltage, ν_(s) is the generated signal, s₂₂ ^(ss)is an impedance of the signal generator, s₂₁, s₁₂, s₃₁, s₂₃, s₃₂, s₃₃are the scatter parameter terms of the power divider, s₁₁ ^(scope) is aninput impedance of the real-time oscilloscope, and s₁₁ ^(DUT) is thereflection coefficient.
 14. The system of claim 13, wherein thecalculated reflection coefficient of the first port of the device undertest is s₁₁ ^(DUT) and the calculated reflection coefficient of thesecond port of the device under test is s₁₁ ^(DUT), and whereinreal-time oscilloscope is further configured to calculate the insertionloss coefficient for each port using the equation:$\frac{b_{2}}{v_{s}} = {\frac{1}{2} \cdot \frac{\left( {1 - s_{22}^{ss}} \right)s_{21}^{DUT}}{1 - {s_{22}^{ss}s_{11}^{DUT}} - {s_{22}^{DUT}s_{11}^{scope}}}}$where s₂₁ ^(DUT) is the insertion loss coefficient.